Wasserstein gradient flow approach to higher order evolution equations
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Transcript of Wasserstein gradient flow approach to higher order evolution equations
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WASSERSTEIN GRADIENT FLOW
APPROACH TO HIGHER ORDER EVOLUTION
EQUATIONS
University of TorontoEhsan Kamalinejad
Joint work with Almut Burchard
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and
of fourth and higher ordernonlinear evolution equation
Existence Uniqueness
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Gradient Flow on a Manifold
Ingredients:
I. Manifold MII. Metric dIII. Energy function E
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Velocity field 𝜕𝑡 𝑋 𝑡=𝑉 𝑡
Steepest Decent 𝑉 𝑡=−𝛻𝐸 (𝑋 𝑡 )
is the gradient Flow of E
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Wasserstein Gradient Flows
• Manifold • Metric
• Energy function
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is the Wasserstein gradient flow
of E if
Continuity Equation 𝜕𝑡 𝜇𝑡+𝛻 . (𝜇𝑡𝑉 𝑡 )=0
Steepest Decent 𝑉 𝑡∈−𝜕𝐸 (𝜇𝑡 )
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PDE reformulated as Gradient Flow
solves PDE
is the gradient flow of
Where
Thin-Film Equation
Dirichlet Energy
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Displacement Convexity
is geodesic between and
-
𝐸 (𝑢𝑠 )≤ (1−𝑠) 𝐸 (𝑢0 )+𝑠𝐸 (𝑢1 )𝐸 (𝑢𝑠 )≤ (1−𝑠) 𝐸 (𝑢0 )+𝑠𝐸 (𝑢1 )− 𝜆2𝑠 (1−𝑠)𝑊 2¿
d2
d s2 E(us)≥ λW 2 ¿
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Wasserstein Gradiet Flow
McCann 1994Displacement
convexsity
Brenier – McCann 1996-2001Structure of the
Wasserstein metric
Otto, Jordan, Kinderlehrer
1998-2001First gradient flow approach to PDEs
De Giorgi – Ambrosio, Savare, Gigli
1993-2008Systematic proofs
based on Minimizing Movement
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Proofs are based on -convexity assumption
for many interesting cases likeDirichlet energy
(Thin-Film Equation)
Fails
Existence, Uniqueness,
Longtime Behavior of many equations has been studied
Stability, and
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To prove that Thin-Film and related equations are well-posed
using Gradient Flow method
Ideas are to
Study the Convexity Along the Flow( depends might change along the flow)
Use the Dissipation of the Energy (convexity on energy sub-levels)
Relaxed
Our Goal
-convexity assumption
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Restricted -convexity
E is restricted -convexat with if such that E is -convex along geodesics connecting any pair of points inside
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Theorem I
E is Restricted -convex at .
Then the Gradient Flow of E starting from
Exists and is Unique at least locally in time.
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Theorem II
The Dirichlet energy is
restricted -convex
on positive measures (on ).
Periodic solutions of the Thin-Film equation exist and are unique on positive data.
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Minimizing Movement is a CONSTRUCTIVE method
Numerical Approximation
Our local existence-uniqueness result extends directly to more classes of energy functionals of the form:
E (u )=∫∑i=1
m
aiubi∨𝜕x
𝑘𝑖u¿2
Higher order equationsQuantum Drift Diffusion Equation
Global Well-posedness when
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THANK YOU.